Math 2.0

Mathematical logic and foundations Theorem

Contrapositive

Builds on ImplicationNegationLogical equivalence

Statement

Let pp and qq be propositions. The implication p→qp \to q is logically equivalent to its contrapositive ¬q→¬p\neg q \to \neg p. p→q≡¬q→¬pp \to q \equiv \neg q \to \neg p

Remarks

The direction from p→qp \to q to ¬q→¬p\neg q \to \neg p is valid in intuitionistic logic. The other direction needs classical logic. This equivalence is the basis of proof by contrapositive. Do not confuse the contrapositive with the converse q→pq \to p.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionContrapositive, statement, no evidence levelContrapositivestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Conjunction definition
  3. Disjunction definition
  4. Implication definition
  5. Negation definition
  6. Biconditional definition
  7. Truth table concept
  8. Propositional logic concept
  9. Tautology definition
  10. Logical equivalence definition
  11. Contrapositive statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Contraposition, lead section
About this page
Tag04T1CT7212
URIm2://04T1CT7212
Aliasescontraposition, law-of-contraposition
Kindstatement, theorem
WritingW0 stub
MSC03B05
StatusThis copy of the site has no build data. Run m2 status 04T1CT7212.
UsesImplication, Negation, Logical equivalence

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